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<p>Linearly constrained nonconvex and nonsmooth composite optimization problems are prevalent in practical scenarios such as inverse problems and statistical learning, and hold significant application value. To simplify the iteration architecture, eliminate the need for checkpoint settings, and attain higher operational efficiency, by integrating symmetric updating techniques, the Bregman distance, and a hybrid gradient estimator, a novel stochastic alternating direction method of multipliers (ADMM) is proposed to solve large-scale linearly constrained nonconvex and nonsmooth composite optimization. The method allows larger step sizes to handle objective functions in either expectation or finite-sum form, thereby effectively reducing the number of iterations. At the same time, the introduction of the Bregman distance significantly simplifies the complexity of solving the subproblems. Unlike previous stochastic ADMMs that rely on double-loop structures, our approach employs a hybrid gradient estimator to achieve single-loop and single-sample updates, and maintains the currently optimal oracle complexity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {O}(\epsilon ^{-3})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>ϵ</mi> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Without relying on the Kurdyka–Łojasiewicz (KL) property and under appropriate mild conditions, we establish global convergence and sublinear convergence rate of the proposed method. Finally, as an application and a numerical experiment, solving a graph-guided fused LASSO problem is given to verify the effectiveness of our algorithm. Furthermore, conclusions on the main results presented in this article are provided, along with a description of future research directions.</p>

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Large-step symmetric hybrid stochastic Bregman-type ADMM for solving constrained nonconvex and nonsmooth composite optimization under no the KL property

  • 期刊论文

  • 发表日期：

    2026年6月26日

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  • Yi-Xin Yang,
  • Heng-You Lan,
  • Lin-Cheng Jiang

摘要

Linearly constrained nonconvex and nonsmooth composite optimization problems are prevalent in practical scenarios such as inverse problems and statistical learning, and hold significant application value. To simplify the iteration architecture, eliminate the need for checkpoint settings, and attain higher operational efficiency, by integrating symmetric updating techniques, the Bregman distance, and a hybrid gradient estimator, a novel stochastic alternating direction method of multipliers (ADMM) is proposed to solve large-scale linearly constrained nonconvex and nonsmooth composite optimization. The method allows larger step sizes to handle objective functions in either expectation or finite-sum form, thereby effectively reducing the number of iterations. At the same time, the introduction of the Bregman distance significantly simplifies the complexity of solving the subproblems. Unlike previous stochastic ADMMs that rely on double-loop structures, our approach employs a hybrid gradient estimator to achieve single-loop and single-sample updates, and maintains the currently optimal oracle complexity \(\mathcal {O}(\epsilon ^{-3})\) O ( ϵ - 3 ) . Without relying on the Kurdyka–Łojasiewicz (KL) property and under appropriate mild conditions, we establish global convergence and sublinear convergence rate of the proposed method. Finally, as an application and a numerical experiment, solving a graph-guided fused LASSO problem is given to verify the effectiveness of our algorithm. Furthermore, conclusions on the main results presented in this article are provided, along with a description of future research directions.